The White House is rushing to come up with an unspecified fix as early as this week to counter the millions of health coverage cancellations going to consumers, at the same time it promises improvements in a federal website so balky that enrollments totaled fewer than 27,000 in 36 states combined.

The White House also is taking a more open approach to changes in the law itself. “We welcome sincere efforts,” presidential press secretary Jay Carney said Wednesday at the White House as Democratic impatience grew over a program likely to be at the center of next year’s midterm elections for control of Congress.

After weeks of highly publicized technical woes, the administration had said in advance the enrollment numbers would fall far short of initial expectations.

They did, easily.

A paltry 26,794 people enrolled for health insurance during the first, flawed month of operations for the federal “Obamacare” website.

Adding in enrollment of more than 79,000 in the 14 states with their own websites, the nationwide number of 106,000 October sign-ups was barely one-fifth of what officials had projected — and a small fraction of the millions who have received private coverage cancellations as a result of the federal law.

The administration said an additional 1 million people have been found eligible to buy coverage in the markets, with about one-third qualifying for tax credits to reduce their premiums. Another 396,000 have been found eligible for Medicaid, which covers low-income people.

Republicans were unmoved.

“Even with the administration’s Enron-like accounting, fewer people have signed up for Obamacare nationwide than the 280,000 who’ve already lost their plan in Kentucky as a result of Obamacare mandates,” Senate Minority Leader Mitch McConnell, R-Ky., said.

Administration officials and senior congressional Democrats expressed confidence in the program’s future. “We expect enrollment will grow substantially throughout the next five months,” said Health and Human Services Secretary Kathleen Sebelius, who is in overall charge.

“Even with the issues we’ve had, the marketplace is working and people are enrolling,” she added.

Despite the expressions, the White House raced to reassure anxious Democrats who are worried about the controversial program, which they voted into existence three years ago over Republican opposition as strong now as it was then.

Senate Democrats arranged a closed-door meeting for midday Thursday in the Capitol with White House officials, who held a similar session Wednesday with the House rank and file.

So far, five Senate Democrats are on record in support of legislation by Sen. Mary Landrieu, D-La., to make sure everyone can keep their present coverage if they want to. The bill would require insurance companies to continue offering existing policies, even if they fall short of minimum coverage requirements in the law.

The measure has little apparent chance at passage, given that it imposes a new mandate on the insurance industry that Republicans will be reluctant to accept.

At the same time, a vote would at least permit Democrats to say they have voted to repair some of the problems associated with the Affordable Care Act, as many appear eager to do.

In a statement, Landrieu said Sens. Jeff Merkley of Oregon, Kay Hagan of North Carolina and Mark Pryor of Arkansas were now supporting the legislation, as is Sen. Dianne Feinstein of California. All but Feinstein are on the ballot next year.

Across the Capitol, majority Republicans in the House set a vote for Friday on legislation to permit insurance companies to continue selling existing policies that have been ordered scrapped because they fall short of coverage standards in the law.

While House passage of the measure is assured, each Democrat will be forced to cast a vote on the future of a program that Republicans have vowed to place at the center of next year’s campaign.

Democratic Rep. Mike Doyle of Pennsylvania, who voted for the initial Obama health care bill, said Thursday that members of his caucus want an opportunity to go on the record in support of allowing people to keep the insurance they had.

Doyle told MSNBC in an interview that at a White House meeting Wednesday, House Democrats told Obama about “the frustration level that many of us have” with the health care roll-out.

Doyle said Democrats warned Obama that “if you don’t give us something by Friday” to fix the insurance cancellation problem, then many Democrats are likely to vote for the pending House bill sponsored by Republican Rep. Fred Upton of Michigan, which would accomplish that goal.

The promise of keeping coverage was Obama’s oft-stated pledge when the legislation was under consideration, a calling card since shredded by the millions of cancellations mailed out by insurers.

Obama apologized last week for the broken promise, but aides said at the time the White House was only considering administration changes, rather than new legislation.